Standard form

Standard form: Standard form is way of expressing numbers that are either very large or too small to use into a simplified form

For example the number;

1,000,000,000,000,000,000,000,000

This is such a very huge number to read or write. It would be incredibly hard to use in any maths calculation and you wouldn’t be able to enter such a number in a calculator. It is also very hard to remember! You would have to find some easy way to write it. This is where a standard form expression is useful. This number can be written as;

1024

As you can see the number is very easy to remember and it would be easy to use in any maths calculation since it has a few digits.

0.00000000000000000000000009

Similarly with this number it is just too small to work with, it has too many numbers of 0s and looks to busy to remember. This number can be written as;

9 × 10-27

The following table offers an overview of the kind of numbers that can be rewritten in standard form.

Standard form

…means

Actual value

2 x 103

2 x 10 x 10 x 10

2000

2 x 102

2 x 10 x 10

200

2 x 101

2 x 10

20

2 x 100

2 x 1

2

2 x 10-1

2 x 0.1

0.2

2 x 10-2

2 x 0.1 x 0.1

0.02

2 x 10-3

2 x 0.1 x 0.1 x 0.1

0.002

Writing intergers in standard form

Example: Convert the following numbers into standard form;

40,000

450,000

Answer:

40,000 = 4 × 104

450,000 = 4.5 × 105

Explanation:

We shall write 40,000 in standard form first. We follow a few basic steps.

Step 1: Find the highest multiple between 1 and 10 but less than 10

Find the highest possible multiple between 1 and 10 but less than 10. It can be a decimal number. In this example the highest multiple is a 4. Or you choose the first integer of the number 40,000. In this case 4.

40,000 = 4 × 10,000

Step 2:Find the number of 10 powers

We have to find the number 10 powers that can be multiplied with 4 to get 40,0000. A number that can be multiplied with 4 to get 40,000 is 1000. Note 10,000 can be coverted into an indice which is much easier to remember.

10,000 = 104

Step 3:Rewrite in standard form.

40,000 can be written in standard form by simply multiplying 4 by the integer.

40,000 = 4 × 104

When writing numbers into standard form you must remember this very important rule;

The first number must always be a number between 1 and less than 10 but not 10.

Next we shall write 450,000 in standard form. Note there is a difference between this number and the previous number. Again we follow the same steps.

Step 1: Find the highest multiple between 1 and 10 but less than 10

Find the highest possible multiple between 1 and 10 but less than 10. This number can be a decimal number. Or we could pick the first non zero integers and create the smallest decimal number possible. You do this by putting the decimal number after the first integer. In this case the number is 4.5

450,000 = 4.5 × 100,000

Step 2: Find the number of 10 powers.

We have to find the number of 10 powers that can be multiplied with 4.5 to get 450,000. In this case 100,000

100,000 = 105

Step 3: Rewrite in standard form

Given what we have found out about the number we can rewrite it in standard form.

450,000 = 4.5 × 105

Writing decimals in standard form

Example: Write 0.0000006 in standard form

Answer:

0.0000006 = 6 × 10-7

Explanation:

To write this number in standard form we follow a few basic steps. Just like we did before.

Step 1: Find the highest multiple between 1 and 10 but less than 10

Find the highest possible multiple between 1 and 10 but less than 10. In this case the number is 6.

0.0000006 = 6 × 0.0000001

Step 2: Find the number of 10 powers.

We must find the number of powers that when multiplied with should provide 0.0000006. The number of powers is 0.0000001. When working with decimals you find this by counting the number of digits after the first decimal. This number of digits will represent a negative number power of 10. In this case there is 7 digits after the decimal point.

0.0000001 = 10-7

Step 3: Rewrite in standard form

We have found 10-7. We can use this as a factor for the highest multiple to rewrite 0.0000006 in standard form.

0.0000006 = 6 × 10-7

Example: Convert 0.000434 to standard form.

Answer:

0.000434 = 4.34 × 10-4

Explanation:

We follow the same steps as we did in the previous example.

Step 1: Find the highest multiple between 1 and 10 but less than 10

Find the highest possible multiple between 1 and 10 but less than 10. In this case the number is 4.34. You have to remember that the first number is always less that 10 (positive)

0.000434 = 4.34 × 0.0001

Step 2:Find the number of 10 powers.

We must find the number of powers that when multiplied with should provide 0.0001.

0.0001 = 10-4

Step 3: Rewrite in standard form

Using what we know about the number we can rewrite it as;

0.000434 = 4.34 × 10-4

You must always remember when working with standard forms the first number is always between 1 and less than 10

For example; 13 x 104 is not in standard form, neither is 0.13 x 10-3. The correct standard form is; 1.3 x 104 or 1.3 x 10-3respectively. Notice the first number is between 1 and less than 10. Writing numbers greater that 10 is a very common mistake to make.